Duality for Constructive Modal Logics: from Sahqlvist to Goldblatt-Thomason

2026-01-07

AI Summary

This study investigates the semantic foundations of constructive modal logic CK, aiming to establish a systematic connection between its algebraic semantics and bi-relational semantics and to resolve the frame definability problem. To this end, the paper constructs, for the first time, a categorical duality framework for CK that unifies these two semantic perspectives. Building on this duality, the classical Sahlqvist correspondence theory and the Goldblatt–Thomason definability theorem from modal logic are extended to the constructive setting. The authors successfully prove Sahlqvist-type correspondence and strong completeness results for CK and provide a Goldblatt–Thomason-style characterization of the classes of frames definable in this logic.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Description Logics
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Security and Privacy: Applications of cryptography

Abstract

We carry out a semantic study of the constructive modal logic CK. We provide a categorical duality linking the algebraic and birelational semantics of the logic. We then use this to prove Sahlqvist style correspondence and completeness results, as well as a Goldblatt-Thomason style theorem on definability of classes of frames.

Problem

• constructive modal logic
• duality
• Sahlqvist correspondence
• Goldblatt-Thomason theorem
• frame definability

Innovation

• categorical duality
• constructive modal logic
• Sahlqvist correspondence
• Goldblatt-Thomason theorem
• birelational semantics